1
Distance and Similarity Measures of Interval
Neutrosophic Soft Sets
Said Broumi1,Irfan Deli 2 and Florentin Smarandache3
1 Faculty of Arts and Humanities, Hay El Baraka Ben M'sik Casablanca B.P. 7951, Hassan II University Mohammedia-Casablanca , Morocco
2Kilis 7 AralΔ±k University, 79000 Kilis, Turkey,
3Department of Mathematics, University of New Mexico,705 Gurley Avenue, Gallup, NM 87301, USA
Abstract:In this paper several distance and similarity measures of interval neutrosophic soft
sets are introduced. The measures are examined based on the geometric model, the set
theoretic approach and the matching function.Finally,we have successfully shown an
application of this similarity measure of interval neutrosophic soft sets.
Keywords: Distance, Similarity Measure, Neutrosophic set, Interval Neutrosohic sets,
Interval Neutrosohic Soft sets.
1. Introdction
In 1965, fuzzy set theory was firstly given by Zadeh [2] which is applied in many real
applications to handle uncertainty.Then,interval-valued fuzzy set [3],intuitionisticfuzzy set
theory[4] and interval valued intuitionistic fuzzy sets[5] was introduced by TΓΌrkΕen,
Atanassov and Atanassov and Gargov, respectively. This theories can only handle incomplete
information not the indeterminate information and inconsistent information which exists
commonly in belief systems. So, Neutrsophic sets, founded by F.Smarandache [1], has
capapility to deal with uncertainty, imprecise, incomplete and inconsistent information which
exist in real world from philosophical point of view. The theory is a powerful tool formal
framework which generalizes the concept of the classic set, fuzzy set [2], interval-valued
fuzzy set [3], intuitionistic fuzzy set [4] interval-valued intuitionistic fuzzy set [5], and so on.
In the actual applications, sometimes, it is not easy to express the truth-membership,
indeterminacy-membership and falsity-membership by crisp value, and they may be easier to
expressed by interval numbers. The neutrosophic set and their operators need to be specified
from scientific or engineering point of view. So, after the pioneering work of Smarandache,
in 2005, Wang [6] proposed the notion of interval neutrosophic set ( INS for short) which is
another extension of neutrosophic sets. INS can be described by a membership interval, a
non-membership interval and indeterminate interval, thus the interval value (INS) has the
virtue of complementing NS, which is more flexible and practical than neutrosophic set. The
sets provides a more reasonable mathematical framework to deal with indeterminate and
inconsistent information.A lot of works about neutrosophic set theory have been studied by
several researches [7,11,13,14,15,16,17,18,19,20 ].
2
In 1999, soft theory was introduced byMolodtsov[45] as a completely new mathematical
toolfor modeling uncertainties. After Molodsov, based on the several operations on softsets
introduced in [33,34,35,36,46],some more properties and algebra may be found in [32,34].
We can found some new conceptc ombined with fuzzy set in [28,29,37,39,42], interval-valued
fuzzy set in [38], intuitionistic fuzzy set in [50], rough set in [43,47], interval-valued
intuitionistic fuzzy set in [45], neutrosophic set in [8,9,27], interval neutrosophic set [31].
Also in some problems it is often needed to compare two sets such as fuzzy, soft,
neutrosophic etc. Therefore, some researchers has studied of similarity measurement between
fuzzysets in [24,48], interval valued fuzzy in [48], neutrosophic set in [23,26], interval
neutrosophic set in [10,12].Recently similarity measure of softsets [40,49], intuitionistic fuzzy
soft sets [30]was studied. Similarity measure between two sets such as fuzzy, soft has been
defined by many authors which are based on both distances and matching function. The
significant differences between similarity measure based on matching function and similarity
measure based on distance is that if intersection of the two sets equals empty, then between
similarity measure based on matching function the two sets is zero in but similarity measure
based on distance may not be equal to zero. Distance-based measures are also popular because
it is easier to calculate the intermediate distance between two fuzzy sets or soft sets. Itβs
mentioned in [40]. In this paper several distance and similarity measures of interval
neutrosophic soft sets are introduced. The measures are examined based on the geometric
model, the set-theoretic approach and the matching function. Finally, we give an
application for similarity measures of interval neutrosophic soft sets
2. Prelimiairies
This section gives a brief overview of concepts of neutrosophic set [1], and interval valued
neutrosophic set [6], soft set [41],neutrosophic soft set [27] and interval valued neutrosophic
soft set [31]. More detailed explanations related to this subsection may be found in
[8,9,27,31,36].
Definition 2.1[ 1]Neutrosophic Sets
Let X be an universe of discourse, with a generic element in X denoted by x, the
neutrosophic (NS) set is an object having the form
A = < x: TA(x), IA(x), FA(x)>,x βX, where the functions T, I, F : Xβ ]β0, 1+[ define
respectively the degree of membership (or Truth) , the degree of indeterminacy, and the
degree of non-membership (or Falsehood) of the element x βX to the set A with the
condition.
β0 β€ TA(x) + IA(x)+ FA(x)β€ 3+. (1)
From philosophical point of view, the neutrosophic set takes the value from real standard or
non-standard subsets of ]β0, 1+[. So instead of ] β0, 1+[ we need to take the interval [0, 1] for
technical applications, because ]β0, 1+[ will be difficult to apply in the real applications such
as in scientific and engineering problems.
3
For two NS ,π΄ππ = <x , TA(x) , IA(x), FA(x)> | x β X (2)
And π΅ππ=<x , TB(x) , IB(x), FB(x)> | x β X the two relations are defined as follows:
(1)π΄ππ β π΅ππif and only if TA(x) β€ TB(x) ,IA(x) β₯ IB(x) ,FA(x) β₯ FB(x)
(2)π΄ππ = π΅ππ if and only if , TA(x) =TB(x) ,IA(x) =IB(x) ,FA(x) =FB(x)
Definition 2.2 [6] Interval Valued Neutrosophic Sets
Let X be a universe of discourse, with generic element in X denoted by x. An interval valued
neutrosophic set (for short IVNS) A in X is characterized by truth-membership function
TA(x), indeteminacy-membership function IA(x) and falsity-membership function FA(x). For
each point x in X, we have that TA(x), IA(x), FA(x) β [ 0 ,1] .
For two IVNS , π΄IVNS = <x , [TAL(x), TA
U(x)] , [IAL(x), IA
U(x)] , [FAL(x), FA
U(x)]> | x β X (3)
And π΅IVNS =<x , [TBL(x), TB
U(x)] , [IBL(x), IB
U(x)] , [FBL(x), FB
U(x)] > | x β X the two relations
are defined as follows:
(1)π΄IVNS β π΅IVNSif and only if TAL(x) β€ TB
L(x),TAU(x) β€ TB
U(x) , IAL(x) β₯ IB
L(x) ,IAU(x) β₯
IBU(x) , FA
L(x) β₯ FBL(x) ,FA
U(x) β₯ FBU(x)
(2)π΄IVNS = π΅IVNS if and only if , TAL(xi) = TB
L(xi) ,TAU(xi) = TB
U(xi) ,IAL(xi) = IB
L(xi) ,
IAU(xi) = IB
U(xi) ,FAL(xi) = FB
L(xi) and FAU(xi) = FB
U(xi) for any x β X.
The complement of π΄IVNS is denoted by π΄πΌππππ and is defined by
π΄πΌππππ =<x , [FA
L(x), FAU(x)]> , [1 β IA
U(x), 1 β IAπΏ (x)] , [TA
L(x), TAU(x)]| x β X
Definition 2.3 [41] Soft Sets
Let U be an initial universe set and E be a set of parameters. Let P(U) denotes the power set of
U. Consider a nonempty set A, A β E. A pair (F, A) is called a soft set over U, where F is a
mapping given by F: A β P (U).
It can be written a set of ordered pairs(F, A) = (x, πΉ (x)): xβA.
As an illustration, let us consider the following example.
Example 1 Suppose that U is the set of houses under consideration, say U = h1, h2, . . ., h5.
Let E be the set of some attributes of such houses, say E = e1, e2, . . ., e6, where e1, e2, . . ., e6
stand for the attributes βexpensiveβ, βbeautifulβ, βwoodenβ, βcheapβ, βmodernβ, and βin bad
repairβ, respectively.
In this case, to define a soft set means to point out expensive houses, beautiful houses, and so
on. For example, the soft set (F,A) that describes the βattractiveness of the housesβ in the
opinion of a buyer, say Thomas, may be defined like this:
A=e1,e2,e3,e4,e5;
F(e1) = h2, h3, h5, F(e2) = h2, h4, F(e3) = h1, F(e4) = U, F(e5) = h3, h5.
4
Definition 2.4 Neutrosophic soft Sets [27 ]
Let U be an initial universe set and E be a set of parameters. Consider AβE. LetN(U) denotes
the set of all neutrosophic sets of U. Thecollection (F,A) is termed to be the soft neutrosophic
set over Udenoted by N, where F is amappinggivenbyF : AβP(U).
It can be written a set of ordered pairs π= (x, πΉ (x)): xβA.
Definition 2.5 Interval Valued Neutrosophic Soft Sets [31]
Let U be an universe set, IVN(U) denotes the set of all interval valued neutrosophic sets of U
and E be a set of parameters that are describe the elements of U. Thecollection (K, E) is
termed to be the interval valued neutrosophic soft sets (ivn-soft sets)over U denoted byΞ₯,
where K is a mapping given by K : EβIVN(U).
It can be written a set of ordered pairs
Ξ₯= (x, πΎ(x)): xβE
Here, πΎ which is interval valued neutrosophic sets, is called approximate function of the ivn-
soft sets Ξ₯ and πΎ(x) is called x-approximate value of x βE.
Generally, K, L, M,... will be used as an approximate functions of Ξ₯, Ξ¨ , Ξ©β¦ respectively.
Note that the sets of all ivn-soft sets over U will be denoted by IVNS(U).
Then a relation form of Ξ₯ is defined by RK= (rK(e,u)/(e, u)) : uβU, eβE
where
rK: ExUβIV NS(U) and rK (ππ,π’π)=πππfor all ππ β E and π’π βU.
Here,
1. Ξ₯ is an ivn-soft subset of Ξ¨, denoted by Ξ₯ β Ξ¨, if K(e) βL(e) for alleβE.
2. Ξ₯ is an ivn-soft equals toΞ¨, denoted by Ξ₯ = Ξ¨, if K(e)=L(e) for all eβE.
3. The complement of Ξ₯ is denoted byΞ₯π , and is defined by Ξ₯π = (x, πΎπ (x)): xβE
As an illustration for ivn-soft, let us consider the following example.
Example 2.Suppose that U is the set of houses under consideration, say U = h1, h2, β3. Let
E be the set of some attributes of such houses, say E = e1, e2, π3, π4, where e1, e2, . . ., e6 stand
for the attributes βexpensiveβ, βbeautifulβ, βwoodenβ, βcheapβ, βmodernβ, and βin bad repairβ,
respectively.
In this case we give an ivn-soft set as;
Ξ₯= (π1,<β1,[0.5, 0.6], [0.6 ,0.7],[0.3,0.4]> ,<β2, [0.5, 0.6], [0.6 ,0.7],[0.3,0.4]> ,
<β3, [0.5, 0.6], [0.6,0.7],[0.3,0.4]> ),(π2,<β1,[0.2, 0.3], [0.5 ,0.6],[0.3,0.6]> ,
<β2, [0.4, 0.6], [0.2 ,0.3],[0.2,0.3]> , <β3, [0.5, 0.6], [0.6 ,0.7],[0.3,0.4]> ),
(π3,<β1,[0.3, 0.4], [0.1 ,0.5],[0.2,0.4]>,<β2, [0.2, 0.5], [0.3,0.4],[0.4,0.5]>,
5
<β3, [0.5, 0.6], [0.6,0.7],[0.3,0.4]> ),(π4,<β1,[0.4, 0.6], [0.3 ,0.5],[0.3,0.4]>,
<β2, [0.4, 0.6], [0.2 ,0.3],[0.2,0.3]>) ,<β3, [0.3, 0.4], [0.2,0.7],[0.1,0.4]>)
Definition 2.6 (Distance axioms)
Let E be a set of parameters. Suppose that Ξ₯ = <K,E>, Ξ¨ = <L,E> and Ξ© = <M,E>; are three
ivn-soft sets in universe U. Assume d is a mapping,
d :IVNS(U) x IVNS(U) βΆ [0, 1].If d satisfies the following properties ((1)-(4)) :
(1) d (Ξ₯, Ξ¨) β₯ 0;
(2) d (Ξ₯, Ξ¨) = d (Ξ¨, Ξ₯) ;
(3) d (Ξ₯, Ξ¨) = 0 iff Ξ¨ = Ξ₯;
(4) d (Ξ₯,Ξ¨) + d (Ξ¨, Ξ©) β₯ d (Ξ₯,Ξ©) .
Hence d(Ξ₯,Ξ¨) is called a distance measure between ivn-soft sets Ξ₯ and Ξ¨.
Definition 2.7(similarity axioms)
A real function S: INS(U) x INS(U) βΆ [0, 1] is named a similarity measure between two ivn-
soft set Ξ₯=(K,E) and Ξ¨ =(M,E) if S satisfies all the following properties:
(1) S (Ξ₯, Ξ¨) β [0, 1];
(2) S(Ξ₯, Ξ₯)=S(Ξ¨, Ξ¨) = 1;
(3) S(Ξ₯, Ξ¨) = S(Ξ¨, Ξ₯);
(4) S (Ξ₯, Ξ©) β€ S (Ξ₯,Ξ¨) and S (Ξ₯, Ξ©) β€ S (Ξ¨,Ξ©) if Ξ₯ β Ξ¨ β Ξ©
Hence S(Ξ₯,Ξ¨) is called a similarity measure between ivn-soft setsΞ₯ and Ξ¨.
For more details on the algebra and operations on interval neutrosophic set and soft set and
interval neutrosophic soft set, the reader may refer to [ 5,6,8,9,12, 31,45,52].
3. Distance Measure between Interval Valued Neutrosophic Soft Sets
In this section, we present the definitions of the Hamming and Euclidean distances between
ivn-soft sets and the similarity measures between ivn-soft sets based on the distances, which
can be used in real scientific and engineering applications.
Based on Hamming distance between two interval neutrosophic set proposed by Ye[12 ] as
follow:
6
D (A,B)=1
6β [|TA
L(xi) β TBL(xi)| + |TA
U(xi) β TBU(xi)| + |IA
L(xi) β IBL(xi)| + |IA
U(xi) βππ=1
IBU(xi)| + |FA
L(xi) β FBL| + |FA
L(xi) β FBU(xi)|]
We extended it to the case of ivn-soft sets as follows:
Definition 3.1 Let Ξ₯ = (K,E) =[πππ] ππ₯π and Ξ¨ = (M,E)=[πππ] ππ₯π be two ivn-soft sets.
K(e) = <x, [TK(e)L (x),TK(e)
U (x)] , [IK(e)L (x),IK(e)
U (x)] , [FK(e)L (x),TK(e)
U (x)]>: x β X
M(e) = <x, [TM(e)L (x),TM(e)
U (x)] , [IM(e)L (x),IM(e)
U (x)] , [FM(e)L (x),FM(e)
U (x)]>: x β X
Then we define the following distances for Ξ₯ and Ξ¨
(1) The Hamming distance ππΌπππππ» (Ξ₯ ,Ξ¨),
ππΌπππππ» (Ξ₯ ,Ξ¨) =β β
[|ΞijL T|+|Ξij
UT|+|ΞijL I|+|Ξij
UI|+|ΞijL F|+|Ξij
UF|]
6
ππ=1
ππ=1
Where ΞijLT= TK(e)
L (xi) β TM(e)L (xi) ,Ξij
UT= TK(e)U (xi) β TM(e)
U (xi) ,ΞijLI= IK(e)
L (xi) β IM(e)L (xi)
,ΞijUI= IK(e)
U (xi) β IM(e)U (xi) ,Ξij
LF= FK(e)L (xi) β FM(e)
L (xi) and ΞijUF= FK(e)
U (xi) β FM(e)U (xi)
(2) The normalized Hamming distance ππΌππππππ» (Ξ₯ ,Ξ¨),
ππΌππππππ» (Ξ₯ ,Ξ¨) =
ππΌπππππ» (Ξ₯ ,Ξ¨)
ππ
(3)The Euclidean distanceππΌπππππΈ (Ξ₯ ,Ξ¨),
ππΌπππππΈ (Ξ₯ ,Ξ¨)=ββ β
(ΞijL T)2+(Ξij
UT)2+(ΞijL I)2+(Ξij
UI)2+(ΞijL F)2+(Ξij
UF)2
6ππ=1
ππ=1
Where ΞijLT= TK(e)
L (xi) β TM(e)L (xi) ,Ξij
UT= TK(e)U (xi) β TM(e)
U (xi) ,ΞijLI= IK(e)
L (xi) β IM(e)L (xi)
,ΞijUI= IK(e)
U (xi) β IM(e)U (xi) ,Ξij
LF= FK(e)L (xi) β FM(e)
L (xi) and ΞijUF= FK(e)
U (xi) β FM(e)U (xi)
(4)The normalized Euclidean distance ππΌππππππΈ (Ξ₯ ,Ξ¨),
ππΌππππππΈ (Ξ₯ ,Ξ¨)=
ππΌπππππΈ (Ξ₯ ,Ξ¨)
βππ
Here, it is clear that the following properties hold:
(1) 0 β€ ππΌπππππ» (Ξ₯ ,Ξ¨) β€ m n and 0 β€ ππΌππππ
ππ» (Ξ₯ ,Ξ¨) β€ 1;
(2) 0 β€ ππΌπππππΈ (Ξ₯ ,Ξ¨) β€ βππ and 0 β€ ππΌππππ
ππΈ (Ξ₯ ,Ξ¨) β€ 1;
Example 3.Assume that two interval neutrosophic soft sets Ξ₯ and Ξ¨ are defined as follows
K (e1) =(<π₯1,[0.5, 0.6],[0.6 ,0.7],[0.3,0.4]>,<π₯2, [0.5, 0.6], [0.6 ,0.7],[0.3,0.4]>),
7
K (e2) =(<π₯1,[0.2, 0.3], [0.5 ,0.6],[0.3,0.6]> ,<π₯2, [0.4, 0.6], [0.2 ,0.3],[0.2,0.3]>),
M (e1) =(<π₯1,[0.3, 0.4], [0.1 ,0.5],[0.2,0.4]> ,<π₯2, [0.2, 0.5], [0.3 ,0.4],[0.4,0.5]>),
M (e2) =(<π₯1,[0.4, 0.6], [0.3 ,0.5],[0.3,0.4]> ,<π₯2, [0.3, 0.4], [0.2,0.7],[0.1,0.4]>),
ππΌπππππ» (Ξ₯ ,Ξ¨) = β β
[|ΞijL T|+|Ξij
UT|+|ΞijL I|+|Ξij
UI|+|ΞijL F|+|Ξij
UF|]
6
2π=1
2π=1
=|0.5β0.3|+|0.6β0.4|+|0.6β0.1|+|0.7β0.5|+|0.3β0.2|+|0.4β0.4|
6
+|0.5β0.2|+|0.6β0.5|+|0.6β0.3|+|0.7β0.4|+|0.3β0.4|+|0.4β0.5|
6+
|0.2β0.4|+|0.3β0.6|+|0.5β0.3|+|0.6β0.5|+|0.3β0.3|+|0.6β0.4|
6+
|0.4β0.3|+|0.6β0.4|+|0.2β0.2|+|0.3β0.7|+|0.2β0.1|+|0.3β0.4|
6
ππΌπππππ» (Ξ₯ ,Ξ¨) =0.71
Theorem 3.2 The functions ππΌπππππ» (Ξ₯ ,Ξ¨) , ππΌππππ
ππ» (Ξ₯ ,Ξ¨) , ππΌπππππΈ (Ξ₯ ,Ξ¨) , ππΌππππ
ππΈ (Ξ₯ ,Ξ¨):
IVNS(U) β π +given by Definition 3.1 respectively are metrics, where π + is the set of all
non-negative real numbers.
Proof. The proof is straightforward.
4. Generalized weighted distance measure between two interval valued nutrosophic
soft sets.
Let A and B be two interval neutrosophic sets, then S.Broumi and F.Smarandache[11]
proposed a generalized interval valued neutrosophic weighted distance measure between A
and B as follows:
ππ(π΄ , π΅) = 1
6βπ
π=1 β π€π[|ππ΄πΏ(π₯π) β ππ΅
πΏ(π₯π)|π + |ππ΄π(π₯π) β ππ΅
π(π₯π)|π + |πΌπ΄πΏ(π₯π) βπ
π=1
πΌπ΅πΏ(π₯π)|π + |πΌπ΄
π(π₯π) β πΌπ΅π(π₯π)|π + |πΉπ΄
πΏ(π₯π) β πΉπ΅πΏ(π₯π)|π + |πΉπ΄
π(π₯π) β πΉπ΅π(π₯π)|π]
1
π (4)
where
π> 0and ππ΄πΏ(π₯π) ,ππ΄
π(π₯π),πΌπ΄πΏ(π₯π) ,πΌπ΄
π(π₯π), ,πΉπ΄πΏ(π₯π) ,πΉπ΄
π(π₯π), ππ΅πΏ(π₯π) ,ππ΅
π(π₯π), πΌπ΅πΏ(π₯π) ,πΌπ΅
π(π₯π),
πΉπ΅πΏ(π₯π) ,πΉπ΅
π(π₯π), β [ 0, 1]
,we extended the above equation (4) distance to the case of interval valued neutrosophic soft
set between Ξ₯ and Ξ¨ as follow:
ππ(Ξ₯ , Ξ¨) = 1
6βπ
π=1 β π€π [|ΞijL T|
π+ |Ξij
UT|π
+ |ΞijL I|
π+ |Ξij
UI|π
+ |ΞijL F|
π+ |Ξij
UF|π
]ππ=1
1
π(5)
Where ΞijLT= TK(e)
L (xi) β TM(e)L (xi) ,Ξij
UT= TK(e)U (xi) β TM(e)
U (xi) ,ΞijLI= IK(e)
L (xi) β IM(e)L (xi)
,ΞijUI= IK(e)
U (xi) β IM(e)U (xi) ,Ξij
LF= FK(e)L (xi) β FM(e)
L (xi) and ΞijUF= FK(e)
U (xi) β FM(e)U (xi).
8
Normalized generalized interval neutrosophic distance is
πππ(Ξ₯, Ξ¨) =
1
6πβπ
π=1 β π€π [|ΞijL T|
π+ |Ξij
UT|π
+ |ΞijL I|
π+ |Ξij
UI|π
+ |ΞijL F|
π+ |Ξij
UF|π
]ππ=1
1
π(6)
If w=1
π,
1
π, β¦ ,
1
π,the Eq. (6) is reduced to the following distances:
ππ(Ξ₯ ,Ξ¨) = 1
6βπ
π=1 β [|ΞijL T|
π+ |Ξij
UT|π
+ |ΞijL I|
π+ |Ξij
UI|π
+ |ΞijL F|
π+ |Ξij
UF|π
]ππ=1
1
π (7)
ππ(Ξ₯ ,Ξ¨) = 1
6πβπ
π=1 β [|ΞijL T|
π+ |Ξij
UT|π
+ |ΞijL I|
π+ |Ξij
UI|π
+ |ΞijL F|
π+ |Ξij
UF|π
]ππ=1
1
π (8)
Particular case
(i) if π =1 then the equation (7), (8) is reduced to the following hamming distance and
normalized hamming distance between interval valued neutrosophic soft set
ππΌπππππ» (Ξ₯ ,Ξ¨) = β β
[|ΞijL T|+|Ξij
UT|+|ΞijL I|+|Ξij
UI|+|ΞijL F|+|Ξij
UF|]
6
ππ=1
ππ=1 (9)
ππΌππππππ» (Ξ₯ ,Ξ¨) =
ππΌπππππ» (Ξ₯ ,Ξ¨)
ππ (10)
(ii) If π =2 then the equation (7) , (8) is reduced to the following Euclidean distance and
normalized Euclidean distance between interval valued neutrosophic soft set
ππΌπππππΈ (Ξ₯ ,Ξ¨) = ββ β
(ΞijL T)2+(Ξij
UT)2+(ΞijL I)2+(Ξij
UI)2+(ΞijL F)2+(Ξij
UF)2
6ππ=1
ππ=1 (11)
ππΌππππππΈ (Ξ₯ ,Ξ¨) =
ππΌπππππΈ (Ξ₯ ,Ξ¨)
βππ (12)
5. Similarity Measures between Interval Valued Neutrosophic Soft Sets
This section proposes several similarity measures of interval neutrosophic soft sets.
It is well known that similarity measures can be generated from distance measures. Therefore,
we may use the proposed distance measures to define similarity measures. Based on the
relationship of similarity measures and distance measures, we can define some similarity
measures between IVNSSs Ξ₯ = (K,E) and Ξ¨ = (M,E) as follows:
5.1. Similarity measure based on the geometric distance model
Now for each ππ βE , K( ππ) and M( ππ) are interval neutrosophic set. To find similarity
between Ξ₯ and Ξ¨. We first find the similarity between K(ππ) and M( ππ).
Based on the distance measures defined above the similarity as follows:
ππΌπππππ» (Ξ₯ ,Ξ¨)=
1
1+ππΌπππππ» (Ξ₯ ,Ξ¨)
and ππΌπππππΈ (Ξ₯ ,Ξ¨)=
1
1+ππΌπππππΈ (Ξ₯ ,Ξ¨)
9
ππΌππππππ» (Ξ₯ ,Ξ¨)=
1
1+ππΌππππππ» (Ξ₯ ,Ξ¨)
and ππΌππππππΈ (Ξ₯ ,Ξ¨)=
1
1+ππΌππππππΈ (Ξ₯ ,Ξ¨)
Example 4 : Based on example 3,then
ππΌπππππ» (Ξ₯ ,Ξ¨)=
1
1+0.71 =
1
1.71 = 0.58
Based on (4), we define the similarity measure between the interval valued neutrosophic soft
sets Ξ₯and Ξ¨ as follows:
SDM(Ξ₯ , Ξ¨)= 1- 1
6nβ [|TK(e)
L (xi) β TM(e)L (xi)|
Ξ»+ |TK(e)
U (xi) β TM(e)U (xi)|
Ξ»+ |IK(e)
L (xi) βni=1
IM(e)L (xi)|
Ξ»+ |IK(e)
U (xi) β IM(e)U (xi)|
Ξ»+ |FK(e)
L (xi) β FM(e)L (xi)|
Ξ»+ |FK(e)
U (xi) β
FM(e)U (xi)|
Ξ»]
1
Ξ» (13)
Where Ξ» > 0 πππ SDM(Ξ₯ , Ξ¨) is the degree of similarity of A and B .
If we take the weight of each element π₯π β X into account, then
SDMw (Ξ₯ , Ξ¨)= 1-
1
6β wi [[|TK(e)
L (xi) β TM(e)L (xi)|
Ξ»+ |TK(e)
U (xi) β TM(e)U (xi)|
Ξ»+n
i=1
|IK(e)L (xi) β IM(e)
L (xi)|Ξ»
+ |IK(e)U (xi) β IM(e)
U (xi)|Ξ»
+ |FK(e)L (xi) β FM(e)
L (xi)|Ξ»
+ |FK(e)U (xi) β
FM(e)U (xi)|
Ξ»]]
1
Ξ» (14)
If each elements has the same importance ,i.e w =1
π,
1
π, β¦ ,
1
π, then (14) reduces to (13)
By definition 2.7 it can easily be known that SDM(Ξ₯ , Ξ¨) satisfies all the properties of
definition..
[|ΞijL T| + |Ξij
UT| + |ΞijL I| + |Ξij
UI| + |ΞijL F| + |Ξij
UF|]
Similarly , we define another similarity measure of Ξ₯ and Ξ¨ as:
S(Ξ₯,Ξ¨) = 1 β
[β (|Ξij
L T|Ξ»
+|ΞijUT|
Ξ»+|Ξij
L I|Ξ»
+|ΞijUI|
Ξ»+|Ξij
L F|Ξ»
+|ΞijUF|
Ξ»)π
π=1
β (|TKL (xi)+TM
L (xi)|Ξ»
+|TKU(xi)+TM
U (xi)|Ξ»
+|IKL (xi)+IM
L (xi)|Ξ»
+|IKU(xi)+IM
U (xi)|Ξ»
+|FKL (xi)+FM
L (xi)|Ξ»
+|FKU(xi)+FM
U (xi)|Ξ»
)ππ=1
]
1
π
(15 )
If we take the weight of each element π₯π β X into account, then
10
S(Ξ₯,Ξ¨) = 1 β
[β wi(|Ξij
LT|Ξ»
+|ΞijUT|
Ξ»+|Ξij
LI|Ξ»
+|ΞijUI|
Ξ»+|Ξij
L F|Ξ»
+|ΞijUF|
Ξ»)π
π=1
β wi(|TKL (xi)+TM
L (xi)|Ξ»
+|TKU(xi)+TM
U (xi)|Ξ»
+|IKL (xi)+IM
L (xi)|Ξ»
+|IKU(xi)+IM
U (xi)|Ξ»
+|FKL (xi)+FM
L (xi)|Ξ»
+|FKU(xi)+FM
U (xi)|Ξ»
)ππ=1
]
1
π
( 16)
This also has been proved that all the properties of definition are satisfied, If each elements
has the same importance, and then (16) reduces to (15)
5.2.Similarity measure based on the interval valuedneutrosophic theoretic approach:
In this section, following the similarity measure between two interval neutrosophic sets
defined by S.Broumi and F.Samarandache in [11], we extend this definition to interval valued
neutrosophic soft sets.
Let ππ(Ξ₯, Ξ¨) indicates the similarity between the interval neutrosophic soft sets Ξ₯ and Ξ¨ .To
find the similarity between Ξ₯ and Ξ¨ first we have to find the similarity between their e -
approximations. Let ππ(Ξ₯, Ξ¨) denote the similarity between the two ππ- approximations K(ππ)
and M(ππ).
Let Ξ₯ and Ξ¨ be two interval valued neutrosophic soft sets, then we define a similarity
measure between K(ππ) and M(ππ) as follows:
ππ(Ξ₯, Ξ¨)=β minTK(ππ)
L (xj),TM(ππ)L (xj)+minTK(ππ)
U (xj),TM(ππ)U (xj) +minIK(ππ)
L (xj),IM(ππ)L (xj)+minIK(ππ)
U (xj),IM(ππ)U (xj)+ minFK(ππ)
L (xj),FM(ππ)L (xj)+minFK(ππ)
U (xj),FM(ππ)U (xj)π
π=1
β maxTK(ππ)L (xj),TM(ππ)
L (xj)+maxTK(ππ)U (xj),TM(ππ)
U (xj) +maxIK(ππ)L (xj),IM(ππ)
L (xj)+maxIK(ππ)U (xj),IM(ππ)
U (xj)+ maxFK(ππ)L (xj),FM(ππ)
L (xj)+maxFK(ππ)U (xj),FM(ππ)
U (xj)ππ=1
(17
))
Then π(Ξ₯, Ξ¨) = maxπ
ππ(Ξ₯, Ξ¨)
The similarity measure has the following proposition
Proposition 4.2
Let Ξ₯ and Ξ¨ be interval valued neutrosophic soft sets then
i. 0 β€ π(Ξ₯, Ξ¨) β€ 1
ii. π(Ξ₯, Ξ¨) =π(Ξ¨, Ξ₯)
iii. π(Ξ₯, Ξ¨) = 1 if Ξ₯= Ξ¨
ππΎ(π)πΏ (π₯π) = ππ(π)
πΏ (π₯π), ππΎ(π)π (π₯π) = ππ(π)
π (π₯π) , πΌπΎ(π)πΏ (π₯π) = πΌπ(π)
πΏ (π₯π), πΌπΎ(π)π (π₯π) =
πΌπ(π)π (π₯π) andπΉπΎ(π)
πΏ (π₯π) = πΉπ(π)πΏ (π₯π) , πΉπΎ(π)
π (π₯π) = πΉπ(π)π (π₯π)for i=1,2,β¦., n
Iv .Ξ₯ β Ξ¨ β Ξ© βS(Ξ₯, Ξ¨) β€ min(S(Ξ₯,Ξ¨), S(Ξ¨,Ξ©)
Proof. Properties (i) and( ii) follows from definition
11
(iii) it is clearly that if Ξ₯ = Ξ¨ β S(Ξ₯, Ξ¨) =1
β β minTK(e)L (xi), TM(e)
L (xi) + minTK(e)U (xi), TM(e)
U (xi) +minIK(e)L (xi), IM(e)
L (xi) +ni=1
minIK(e)U (xi), IM(e)
U (xi) + minFK(e)L (xi), FM(e)
L (xi) + minFK(e)U (xi), FM(e)
U (xi)
=β maxTK(e)L (xi), TM(e)
L (xi) + maxTK(e)U (xi), TM(e)
U (xi) +maxIK(e)L (xi), IM(e)
L (xi) +ni=1
maxIK(e)U (xi), IM(e)
U (xi) + maxFK(e)L (xi), FM(e)
L (xi) + maxFK(e)U (xi), FM(e)
U (xi)
β β [minTK(e)L (xi), TM(e)
L (xi) β maxTK(e)L (xi), TM(e)
L (xi)] +ππ=1
[minTK(e)U (xi), TM(e)
U (xi) βmaxTK(e)U (xi), TM(e)
U (xi)] + [minIK(e)L (xi), IM(e)
L (xi) β
maxIK(e)L (xi), IM(e)
L (xi)] + [minIK(e)U (xi), IM(e)
U (xi) β
maxIK(e)U (xi), IM(e)
U (xi)] + [minFK(e)L (xi), FM(e)
L (xi) β maxFK(e)L (xi), FM(e)
L (xi)] +
[minFK(e)U (xi), FM(e)
U (xi) β maxFK(e)U (xi), FM(e)
U (xi)] =0
Thus for each x,
[minTK(e)L (xi), TM(e)
L (xi) β maxTK(e)L (xi), TM(e)
L (xi)] =0
[minTK(e)U (xi), TM(e)
U (xi) β maxTK(e)U (xi), TM(e)
U (xi)] = 0
[minIK(e)L (xi), IM(e)
L (xi) β maxIK(e)L (xi), IM(e)
L (xi)] =0
[minIK(e)U (xi), IM(e)
U (xi) β maxIK(e)U (xi), IM(e)
U (xi)] =0
[minFK(e)L (xi), FM(e)
L (xi) β maxFK(e)L (xi), FM(e)
L (xi)] =0
[minFK(e)U (xi), FM(e)
U (xi) β maxFK(e)U (xi), FM(e)
U (xi)]=0 holds
Thus TK(e)L (xi) = TM(e)
L (xi) ,TK(e)U (xi) = TM(e)
U (xi) ,IK(e)L (xi) = IM(e)
L (xi) , IK(e)U (xi) =
IM(e)U (xi) ,FA
L(xi) = FBL(xi) and FK(e)
U (xi) = FM(e)U (xi) β Ξ₯ = Ξ¨
(iv) now we prove the last result.
Let Ξ₯ β Ξ¨ β Ξ©,then we have
TK(e)L (xi) β€ TM(e)
L (xi) β€ TCL(x) , TK(e)
U (xi) β€ TM(e)U (xi) β€ TC
L(x) , IK(e)L (x) β₯ IM(e)
L (x) β₯
ICL(x),IA
U(x) β₯ IM(e)U (x) β₯ IC
U(x), FK(e)L (x) β₯ FM(e)
L (x) β₯ FCL(x) ,FK(e)
U (x) β₯ FM(e)U (x) β₯ FC
U(x)
for all x β X .Now
TKL(x) +TK
U(x) +IKL(x) +IK
U(x) +FML (x)+FM
U (x) β₯ TKL(x) +TK
U(x) +IKL(x) +IK
U(x)+FCL(x)+FC
U(x)
And
TML (x) +TM
U(x) +IML (x) +IM
U (x) +FKL(x)+FK
U(x) β₯ TCL(x) +TC
U(x) +ICL(x) +IC
U(x)+FKL(x)+FK
U(x)
12
S(Ξ₯,Ξ¨) =TK
L (x) +TKU(x) +IK
L (x) +IKU(x) +FM
L (x)+FMU (x)
TML (x) +TM
U (x) +IML (x) +IM
U (x) +FKL (x)+FK
U(x)β₯
TkL(x) +TK
U(x) +IKL (x) +IK
U(x)+FCL(x)+FC
U(x)
TCL(x) +TC
U(x) +ICL(x) +IC
U(x)+FKL (x)+FK
U(x) = S(Ξ₯,Ξ©)
Again similarly we have
TML (x) +TM
U(x) +IML (x) +IM
U (x)+FCL(x)+FC
U(x) β₯ TKL(x) +TK
U(x) +IKL(x) +IK
U(x)+FCL(x)+FC
U(x)
TCL(x) +TC
U(x) +ICL(x) +IC
U(x)+FKL(x)+FK
U(x) β₯ TCL(x) +TC
U(x) +ICL(x) +IC
U(x)+FML (x)+FM
U (x)
S(Ξ¨,Ξ©) =TM
L (x) +TMU (x) +IM
L (x) +IMU (x)+FC
L(x)+FCU(x)
TCL(x) +TC
U(x) +ICL(x) +IC
U(x)+FML (x)+FM
U (x)β₯
TKL (x) +TK
U(x) +IKL (x) +IK
U(x)+FCL(x)+FC
U(x)
TCL(x) +TC
U(x) +ICL(x) +IC
U(x)+FKL (x)+FK
U(x) = S(Ξ₯,Ξ©)
βS(Ξ₯,Ξ©) β€ min (S(Ξ₯,Ξ¨) , S(Ξ¨,Ξ©))
Hence the proof of this proposition
If we take the weight of each element π₯π β X into account, then
π(Ξ₯, Ξ¨)= β π€πminTA
L (xi),TBL (xi)+minTA
U(xi),TBU(xi) +minIA
L (xi),IBL (xi)+minIA
U(xi),IBU(xi)+ minFA
L (xi),FBL (xi)+minFA
U(xi),FBU(xi)π
π=1
β π€πmaxTAL (xi),TB
L (xi)+maxTAU(xi),TB
U(xi) +maxIAL (xi),IB
L (xi)+maxIAU(xi),IB
U(xi)+ maxFAL (xi),FB
L (xi)+maxFAU(xi),FB
U(xi)ππ=1
(18)
Particularly ,if each element has the same importance, then (18) is reduced to (17) ,clearly
this also satisfies all the properties of definition .
Theorem Ξ₯ = <K,E>, Ξ¨ = <L,E> and Ξ© = <M,E>; are three ivn-soft sets in universe U such
that Ξ₯is a ivn-soft subset of Ξ¨ and Ξ¨is a soft subset of Ξ© then, S(Ξ₯, Ξ©) β€ S(Ξ¨, Ξ©).
Proof. The proof is straightforward.
5.3. Similarity measure based for matching function by using interval neutrosophic
sets:
Chen [24] and Chen et al. [25]) introduced a matching function to calculate the degree of
similarity between fuzzy sets. In the following, we extend the matching function to deal
with the similarity measure of interval valued neutrosophic soft sets.
Let Ξ₯ = A and Ξ¨=B be two interval valued neutrosophic soft sets, then we define a
similarity measure between Ξ₯ and Ξ¨ as follows:
πππΉ(Ξ₯ ,Ξ¨)=
β ((ππ΄πΏ(π₯π) β ππ΅
πΏ(π₯π)) + (ππ΄π(π₯π) β ππ΅
π(π₯π)) + (πΌπ΄πΏ(π₯π) β πΌπ΅
πΏ(π₯π)) + (πΌπ΄π(π₯π) β πΌπ΅
π(π₯π)) + (πΉπ΄πΏ(π₯π) β πΉπ΅
πΏ(π₯π)) + (πΉπ΄π(π₯π) β πΉπ΅
π(π₯π)))ππ=π
max( β (TAL(xi)
2 + TAU(xi)
2 + IAL(xi)
2 +ππ= IA
U(xi)2 + FA
L(xi)2 + FA
U(xi)2), β (TB
L(xi)2 + TB
U(xi)2 + IB
L(xi)2 +π
π= IBU(xi)
2 + FBL(xi)
2 + FBU(xi)
2))
ππΎ(π)πΏ (π₯π) = ππ(π)
πΏ (π₯π), ππΎ(π)π (π₯π) = ππ(π)
π (π₯π) , πΌπΎ(π)πΏ (π₯π) = πΌπ(π)
πΏ (π₯π), πΌπΎ(π)π (π₯π) =
πΌπ(π)π (π₯π) and πΉπΎ(π)
πΏ (π₯π) = πΉπ(π)πΏ (π₯π) , πΉπΎ(π)
π (π₯π) = πΉπ(π)π (π₯π)
(19)
Proof.
13
i. 0 β€ πππΉ(Ξ₯ ,Ξ¨) β€ 1
The inequality πππΉ(Ξ₯ ,Ξ¨) 0 is obvious. Thus, we only prove the inequality S(Ξ₯ ,Ξ¨) 1.
πππΉ(Ξ₯,Ξ¨)=β ((TK(e)L (xi) β ππ(π)
πΏ (xi)) + (TK(e)U (xi) β ππ(π)
π (xi)) + (πΌπΎ(π)πΏ (π₯π) βπ§
π’=π
πΌπ(π)πΏ (xi)) + (πΌπΎ(π)
π (π₯π) β πΌπ(π)π (xi)) + (πΉπΎ(π)
πΏ (π₯π) β πΉπ(π)πΏ (xi)) + (πΉπΎ(π)
π (π₯π) β πΉπ(π)π (xi)))
=ππΎ(π)πΏ (π₯1) β ππ(π)
πΏ (π₯1)+ππΎ(π)πΏ (π₯2) β ππ(π)
πΏ (π₯2)+β¦+ππΎ(π)πΏ (π₯π) β ππ(π)
πΏ (π₯π)+ππΎ(π)π (π₯1) β
ππ(π)π (π₯1)+ππΎ(π)
π (π₯2) β ππ(π)π (π₯2)+β¦+ππΎ(π)
π (π₯π) β ππ(π)π (π₯π)+
πΌπΎ(π)πΏ (π₯1) β πΌπ(π)
πΏ (π₯1)+πΌπΎ(π)πΏ (π₯2) β πΌπ(π)
πΏ (π₯2)+β¦+πΌπΎ(π)πΏ (π₯π) β πΌπ(π)
πΏ (π₯π)+πΌπΎ(π)π (π₯1) β
πΌπ(π)π (π₯1)+πΌπΎ(π)
π (π₯2) β πΌπ(π)π (π₯2)+β¦+πΌπΎ(π)
π (π₯π) β πΌπ(π)π (π₯π)+
πΉπΎ(π)πΏ (π₯1) β πΉπ(π)
πΏ (π₯1)+πΉπΎ(π)πΏ (π₯2) β πΉπ(π)
πΏ (π₯2)+β¦+πΉπΎ(π)πΏ (π₯π) β πΉπ(π)
πΏ (π₯π)+πΉπΎ(π)π (π₯1) β
πΉπ(π)π (π₯1)+πΉπΎ(π)
π (π₯2) β πΉπ(π)π (π₯2)+β¦+πΉπΎ(π)
π (π₯π) β πΉπ(π)π (π₯π)+
According to the CauchyβSchwarz inequality:
(π₯1 β π¦1 + π₯2 β π¦2 + β― + π₯π β π¦π)2 β€ (π₯12 + π₯2
2 + β― + π₯π2) β (π¦1
2 + π¦22 + β― + π¦π
2)
where (π₯1, π₯2, β¦, π₯π) β π πand (π¦1, π¦2, β¦, π¦π) β π π we can obtain
[πππΉ(πΆ, πΉ)]2 β€ β(ππΎ(π)πΏ (π₯π)
2 + ππΎ(π)π (π₯π)
2 + πΌπΎ(π)πΏ (π₯π)
2 + πΌπΎ(π)π (π₯π)2 + πΉπΎ(π)
πΏ (π₯π)2
π
π=π
+ πΉπΎ(π)π (π₯π)
2) β
β (ππ(π)πΏ (π₯π)
2 + ππ(π)π (π₯π)
2 + πΌπ(π)πΏ (π₯π)
2 + πΌπ(π)π (π₯π)
2 + πΉπ(π)πΏ (π₯π)2 +π
π=π
πΉπ(π)π (π₯π)
2)=S(Ξ₯, Ξ₯)βS(Ξ¨, Ξ¨)
Thus πππΉ(Ξ₯,Ξ¨)β€ [π(Ξ₯, Ξ₯)]1
2 β [π(Ξ¨, Ξ¨)]1
2
Then πππΉ(Ξ₯,Ξ¨)β€maxS(Ξ₯, Ξ₯), S(Ξ¨, Ξ¨)]
Therefore,πππΉ(Ξ₯,Ξ¨)β€ 1.
If we take the weight of each element π₯π β X into account, then
πππΉπ€ (Ξ₯,Ξ¨)=
β π€π ((ππ΄πΏ(π₯π) β ππ΅
πΏ(π₯π)) + (ππ΄π(π₯π) β ππ΅
π(π₯π)) + (πΌπ΄πΏ(π₯π) β πΌπ΅
πΏ(π₯π)) + (πΌπ΄π(π₯π) β πΌπ΅
π(π₯π)) + (πΉπ΄πΏ(π₯π) β πΉπ΅
πΏ(π₯π)) + (πΉπ΄π(π₯π) β πΉπ΅
π(π₯π)))ππ=π
max( β π€π (TAL(xi)
2 + TAU(xi)
2 + IAL(xi)
2 +ππ= IA
U(xi)2 + FA
L(xi)2 + FA
U(xi)2), β π€π (TB
L(xi)2 + TB
U(xi)2 + IB
L(xi)2 +π
π= IBU(xi)
2 + FBL(xi)
2 + FBU(xi)
2))
(20)
Particularly ,if each element has the same importance, then (20) is reduced to (19) ,clearly
this also satisfies all the properties of definition .
14
The larger the value of S(Ξ₯,Ξ¨) ,the more the similarity between Ξ₯ and Ξ¨.
Majumdar and Samanta [40] compared the properties of the two measures of soft sets and
proposed Ξ±-similar of two soft sets. In the following, we extend to interval valued
neutrosophic soft sets as;
Let πΞ₯,Ξ¨ denote the similarity measure between two ivn-soft sets Ξ₯ and Ξ¨ Table compares
the properties of the two measures of similarity of ivn-soft soft sets discussed here. It can be
seen that most of the properties are common to both.and few differences between them do
exist.
Property S (geometric ) π (theoretic) π ( matching)
S(Ξ₯, Ξ¨) = S(Ξ¨, Ξ₯)
0 β€S(Ξ₯, Ξ¨) β€1
Ξ₯ = Ξ¨βS(Ξ₯, Ξ¨) = 1
S(Ξ₯, Ξ¨) = 1 βΞ₯ = Ξ¨
Ξ₯ β© Ξ¨ = β βS(Ξ₯, Ξ¨) = 0
S(Ξ₯, Ξ₯π) = 0
Yes
Yes
Yes
Yes
No
No
Yes
Yes
No
Yes
No
No
Yes
Yes
?
?
?
?
Definition A relation Ξ±β on IVNS(U), called Ξ±-similar, as follows:two inv-soft sets Ξ₯ and Ξ¨
are said to be Ξ±-similar, denoted as Ξ₯Ξ±βΞ¨ iff S(Ξ₯, Ξ¨) β₯ Ξ± for Ξ± β(0, 1).
Here, we call the two ivn-soft sets significantly similar if S(Ξ₯, Ξ¨) >0.5
Lemma [40] Ξ±βis reflexive and symmetric, but not transitive.
Majumdar and Samanta [40] introduced a technique of similarity measure of two soft sets
which can be applied to detect whether an ill person is suffering from a certain disease or not.
In a example, they was tried to estimate the possibility that an ill person having certain visible
symptoms is suffering from pneumonia.Therefore, they were given an example by using
similarity measure of two soft sets. In the following application, similarly we will try for ivn-
soft sets in same example. Some of it is quoted from [40] .
6. An Application
This technique of similarity measure of two inv-soft sets can be applied to detect whether an
ill person is suffering from a certain disease or not. In the followinge xample, we will try to
estimate the possibility that an ill person having certain visible symptoms is suffering from
pneumonia. For this, we first construct a model inv-soft set for pneumonia and the inv-soft set
for the ill person. Next we find the similarity measure of these two sets. If they are
significantly similar, then we conclude that the person is possibly suffering from pneumonia.
Let our universal set contain only two elements yes and no, i.e. U = yes=h1, no=h2. Here
the set of parameters E is the set of certain visible symptoms. Let E = e1, e2, e3, e4, e5, e6,
where e1 = high body temperature, e2 = cough with chest congestion, e3 = body ache, e4 =
headache, e5 = loose motion, and e6 = breathing trouble. Our model inv-soft for
pneumoniaΞ₯is given below and this can be prepared with the help of a medical person:
15
Ξ₯= (π1,<β1,[0.5, 0.6], [0.6 ,0.7],[0.3,0.4]> ,<β2, [0.5, 0.6], [0.6 ,0.7],[0.3,0.4]>),
(π2,< β1, [0.5, 0.6], [0.6,0.7],[0.3,0.4]>,<β2,[0.2, 0.3], [0.5 ,0.6],[0.3,0.6]>),
(π3,<β1, [0.4, 0.6], [0.2 ,0.3],[0.2,0.3]> , <β2,[0.5, 0.6], [0.6 ,0.7],[0.3,0.4]> ),
(π4,<β1,[0.3, 0.4], [0.1 ,0.5],[0.2,0.4]>,<β2, [0.2, 0.5],[0.3,0.4],[0.4,0.5]> ),
(π5,<β1,[0.5, 0.6],[0 .6,0.7],[0.3,0.4]>,<β2,[0.4, 0.6], [0.3 ,0.5],[0.3,0.4]>),
(π6,<β1, [0.4, 0.6], [0.2 ,0.3],[0.2,0.3]>) , <β2, [0.3, 0.4], [0.2,0.7],[0.1,0.4]>)
Now the ill person is having fever, cough and headache. After talking to him, we can
construct his ivn-soft Ξ¨ as follows:
Ξ¨ = (π1,<β1,[0.1, 0.2], [0.1 ,0.2],[0.8,0.9]> ,<β2, [0.1, 0.2], [0.0 ,0.1],[0.8,0.9]>),
(π2,< β1, [0.8, 0.9], [0.1,0.2],[0.2,0.9]> ,<β2,[0.8, 0.9], [0.2 ,0.9],[0.8,0.9]>),
(π3,<β1,[0.1, 0.9], [0.7 ,0.8],[0.6,0.9]> , <β2, [0.1, 0.8], [0.6 ,0.7],[0.8,0.7]> ),
(π4,<β1,[0.8, 0.8], [0.1 ,0.9],[0.3,0.3]>,<β2, [0.6, 0.9],[0.5,0.9],[0.8,0.9]> ),
(π5,<β1, [0.3, 0.4],[0 .1,0.2],[0.8,0.8]> ,<β2,[0.5, 0.9], [0.8 ,0.9],[0.1,0.2]>),
(π6,<β1, [0.1, 0.2], [0.8 ,0.9],[0.7,0.7]>) , <β2, [0.7, 0.8], [0.8,0.9],[0.0,0.4]>)
Then we find the similarity measure of these two ivn-soft sets as:
ππΌπππππ» (Ξ₯ ,Ξ¨)=
1
1+ππΌπππππ» (Ξ₯ ,Ξ¨)
=0.17
Hence the two ivn-softsets, i.e. two symptoms Ξ₯ and Ξ¨are not significantly similar. Therefore,
we conclude that the person is not possibly suffering from pneumonia. A person suffering
from the following symptoms whose corresponding ivn-soft set Ξ© is given below:
Ξ©= (π1,<β1,[0.5, 0.7], [0.5 ,0.7],[0.3,0.5]> ,<β2, [0.6, 0.6], [0.6 ,0.8],[0.3,0.5]>),
(π2,< β1, [0.5, 0.7], [0.5,0.7],[0.3,0.4]> ,<β2,[0.2, 0.4], [0.6 ,0.7],[0.2,0.7]>),
(π3,<β1, [0.4, 0.7], [0.2 ,0.2],[0.1,0.3]> , <β2, [0.4, 0.8], [0.2 ,0.8],[0.2,0.8]> ),
(π4,<β1,[0.3, 0.4], [0.1 ,0.5],[0.2,0.6]>,<β2, [0.2, 0.5],[0.3,0.4],[0.4,0.5]> ),
(π5,<β1, [0.5, 0.6],[0 .6,0.7],[0.3,0.4]> ,<β2,[0.4, 0.6], [0.3 ,0.5],[0.1,0.8]>),
(π6,<β1, [0.4, 0.7], [0.3 ,0.7],[0.2,0.8]>) , <β2, [0.5, 0.2], [0.3,0.5],[0.2,0.5]>)
Then,
ππΌπππππ» (Ξ₯ ,Ξ©)=
1
1+ππΌπππππ» (Ξ₯ ,Ξ¨)
= 0.512
Here the two ivn-soft sets, i.e. two symptoms Ξ₯ and Ξ© are significantly similar. Therefore, we
conclude that the person is possibly suffering from pneumonia. This is only a simple example
16
to show the possibility of using this method for diagnosis of diseases which could be
improved by incorporating clinical results and other competing diagnosis.
Conclusions
In this paper we have defined, for the first time, the notion of distance and similarity measures
between two interval neutrosophic soft sets. We have studied few properties of distance and
similarity measures. The similarity measures have natural applications in the field of pattern
recognition, feature extraction, region extraction, image processing, coding theory etc. The
results of the proposed similarity measure and existing similarity measure are compared. We
also give an application for similarity measures of interval neutrosophic soft sets.
Acknowledgements
The authors are very grateful to the anonymous referees for their insightful and constructive
comments and suggestions, which have been very helpful in improving the paper.
References
[1] F.Smarandache, βA Unifying Field in Logics. Neutrosophy: Neutrosophic Probability, Set and Logicβ. Rehoboth: American Research Press,(1998).
[2] L. A. Zadeh, ,"Fuzzy sets". Information and Control, 8,(1965),pp. 338-353.
[3] Turksen, βInterval valued fuzzy sets based on normal formsβ.Fuzzy Sets and Systems, 20, (1968), pp.191β210.
[4] Atanassov, K. βIntuitionistic fuzzy setsβ.Fuzzy Sets and Systems, 20, (1986), pp.87β96.
[5] Atanassov, K., & Gargov, G βInterval valued intuitionistic fuzzy setsβ. Fuzzy Sets and Systems.International Journal of General Systems 393, 31,(1998), pp 343β349
[6] Wang, H., Smarandache, F., Zhang, Y.-Q. and Sunderraman, R.,βIntervalNeutrosophic Sets and Logic:
Theory and Applications in Computingβ, Hexis, Phoenix, AZ, (2005).
[7] A. Kharal, βA NeutrosophicMulticriteria Decision Making Methodβ,New Mathematics & Natural Computation, to appear in Nov 2013
[8] S.Broumi and F. Smarandache, βIntuitionistic Neutrosophic Soft Setβ, Journal of Information and Computing Science, England, UK ,ISSN 1746-7659,Vol. 8, No. 2, (2013), 130-140.
[9] S.Broumi, βGeneralized Neutrosophic Soft Setβ, International Journal of Computer Science, Engineering and Information Technology (IJCSEIT), ISSN: 2231-3605, E-ISSN : 2231-3117, Vol.3, No.2, (2013) 17-30.
[10] S.Broumi and F. Smarandache,β Cosine similarity measure of interval valued neutrosophic setsβ, 2014 (submitted).
[11] S.Broumi and F. Smarandache,β New Distance and Similarity Measures of Interval Neutrosophic Setsβ, 2014 (submitted).
[12] J. Ye,β Similarity measures between interval neutrosophic sets and their multicriteria decision-making method βJournal of Intelligent & Fuzzy Systems, DOI: 10.3233/IFS-120724 ,(2013),.
[13] M.Arora, R.Biswas,U.S.Pandy, βNeutrosophic Relational Database Decompositionβ, International Journal of Advanced Computer Science and Applications, Vol. 2, No. 8, (2011) 121-125.
[14] M. Arora and R. Biswas,β Deployment of Neutrosophic technology to retrieve answers for queries posed in natural languageβ, in 3rdInternational Conference on Computer Science and Information Technology ICCSIT, IEEE catalog Number CFP1057E-art,Vol.3, ISBN: 978-1-4244-5540-9,(2010) 435-439.
[15] Ansari, Biswas, Aggarwal,βProposal for Applicability of Neutrosophic Set Theory in Medical AIβ, International Journal of Computer Applications (0975 β 8887),Vo 27β No.5, (2011) 5-11.
17
[16] F.G LupiÑñez, "On neutrosophic topology", Kybernetes, Vol. 37 Iss: 6,(2008), pp.797 - 800 ,Doi:10.1108/03684920810876990.
[17] S. Aggarwal, R. Biswas,A.Q.Ansari,βNeutrosophic Modeling and Controlβ,978-1-4244-9034-/10 IEEE,( 2010) 718-723.
[18] H. D. Cheng,& Y Guo. βA new neutrosophic approach to image thresholdingβ. New Mathematics and Natural Computation, 4(3), (2008) 291β308.
[19] Y.Guo,&, H. D. Cheng βNew neutrosophic approach to image segmentationβ.Pattern Recognition, 42, (2009) 587β595.
[20] M.Zhang, L.Zhang, and H.D.Cheng. βA neutrosophic approach to image segmentation based on watershed methodβ. Signal Processing 5, 90 , (2010) 1510-1517.
[21] Wang, H., Smarandache, F., Zhang, Y. Q., Sunderraman,R,βSingle valued neutrosophicβ,sets.Multispace and Multistructure, 4,(2010) 410β413.
[22] S. Broumi, F. Smarandache , βCorrelation Coefficient of Interval Neutrosophic setβ, Periodical of Applied Mechanics and Materials, Vol. 436, 2013, with the title Engineering Decisions and Scientific Research in Aerospace, Robotics, Biomechanics, Mechanical Engineering and Manufacturing; Proceedings of the International Conference ICMERA, Bucharest, October 2013.
[23] P. Majumdar, S.K. Samant,β On similarity and entropy of neutrosophicsetsβ,Journal of Intelligent and Fuzzy Systems,1064-1246(Print)-1875-8967(Online),(2013),DOI:10.3233/IFS-130810, IOSPress.
[24] S.M. Chen, S. M. Yeh and P.H. Hsiao, βA comparison of similarity measure of fuzzy valuesβ Fuzzy sets and systems, 72, 1995. 79-89.
[25] S.M. Chen, βA new approach to handling fuzzy decision making problemβ IEEE transactions on Systems, man and cybernetics, 18, 1988. , 1012-1016
[26] Said Broumi, and FlorentinSmarandache, Several Similarity Measures of Neutrosophic Setsβ ,Neutrosophic Sets and Systems,VOL1 ,2013,54-62.
[27] P.K. Maji, Neutrosophic soft set, Computers and Mathematics with Applications, 45 (2013) 555-562.
[28] ΓaΔman, N. Deli, I. Means of FP-SoftSetsandits Applications, Hacettepe Journal of MathematicsandStatistics, 41 (5) (2012), 615β625.
[29] ΓaΔman, N. Deli, I. Product of FP-SoftSetsandits Applications, Hacettepe Journal of MathematicsandStatistics, 41 (3) (2012), 365 - 374.
[30] ΓaΔman, N. Similarity measures of intuitionistic fuzzy soft sets and their decision making,
http://arxiv.org/abs/1301.0456
[31] Deli, I.Interval-valued neutrosophic soft sets and its decision making http://arxiv.org/abs/1402.3130
[32] AktaΕ, H. and ΓaΔman, N., Soft sets and soft groups. Information Sciences, (2007),177(1), 2726-2735.
[33] Babitha, K. V. and Sunil, J. J., Soft set relations and functions. Computers and Mathematics with Applications, 60, (2010), 1840-1849.
[34] Chen, D., Tsang, E.C.C., Yeung, D.S., 2003. Some notes on the parameterization reduction of soft sets. International Conference on Machine Learning and Cybernetics, 3, 1442-1445.
[35] Chen, D., Tsang, E.C.C., Yeung, D.S. and Wang, X., The parameterization reduction of soft sets and its applications. Computers and Mathematics with Applications, 49(1), (2005), 757-763.
[36] ΓaΔman, N., and EnginoΔlu, S., Soft set theory and uni-int decision making. European Journal of Operational Research, 207, (2010b), 848-855.
[37] Feng, F., Jun, Y. B., Liu, X. and Li, L., An adjustable approach to fuzzy soft set based decision making. Journal of Computational and Applied Mathematics, (2010), 234, 10-20.
[38] Feng, F., Li, Y. and Leoreanu-Fotea, V., 2010. Application of level soft sets in decision making based on interval-valued fuzzy soft sets. Computers and Mathematics with Applications, 60, 1756-1767.
[39] Maji, P.K., Biswas, R. and Roy, A.R., Fuzzy soft sets. Journal of Fuzzy Mathematics, (2001) 9(3), 589-602.
[40] Majumdar, P. and Samanta, S. K., Similarity measure of soft sets. New Mathematics and Natural
18
Computation, (2008), 4(1), 1-12.
[41] Molodtsov, D., Soft set theory-first results. Computers and Mathematics with Applications, 37(1), (1999), 19-31.
[42] Roy, A.R. and Maji, P.K., A fuzzy soft set theoretic approach to decision making problems. Journal of Computational and Applied Mathematics, 203(1), (2007), 412-418.
[43] Yang, H., Qu, C., Li, N-C., 2004. The induction and decision analysis of clinical diagnosis based on rough sets and soft sets. (FangzhiGaoxiaoJichukexueXuebao Ed.), September, (2007), 17(3), 208-212.
[44] U. Acar, F. Koyuncu and B. Tanay, Soft sets and soft rings, Computers and Mathematics with Applications, 59, (2010) 3458-3463.
[45] Y. Jiang, Y. Tang, Q. Chen, H. Liu, J.Tang, Interval-valued intuitionistic fuzzy soft sets and their properties, Computers and Mathematics with Applications, 60 (2010) 906-918.
[46] M.I. Ali, F. Feng, X. Liu, W.K. Min and M. Shabir, On some new operations in soft set theory, Computers and Mathematics with Applications, 57, (2009), 1547-1553.
[47] M.I. Ali, A note on soft sets,rough soft sets and fuzzy soft sets, Applied Soft Computing, 11, (2011), 3329-
3332.
[48] P Grzegorzewski, Distances between intuitionistic fuzzy sets and/or interval-valued fuzzy sets based on th
eHausdorff metric, FSS 148,(2004), 319-328.
[49] A. Kharal, Distance and Similarity Measures for Soft Sets, New Math. And Nat. Computation, 06, 321
(2010). DOI: 10.1142/S1793005710001724. [50] P.K. Maji,A.R. Roy, R. Biswas, On Intuitionistic Fuzzy Soft Sets. The Journal of Fuzzy Mathematics,
12(3) (2004) 669-683.